ε-Monotone Fourier Methods for Optimal Stochastic Control in1

Finance2

Peter A. Forsyth ∗ George Labahn†3

Revised: April 3, 20184

Abstract5

Stochastic control problems in finance often involve complex controls at discrete times. As a6

result numerically solving such problems, for example using methods based on partial differen-7

tial or integro-differential equations, inevitably give rise to low order accuracy, usually at most8

second order. In many cases one can make use of Fourier methods to efficiently advance solu-9

tions between control monitoring dates and then apply numerical optimization methods across10

decision times. However Fourier methods are not monotone and as a result give rise to possible11

violations of arbitrage inequalities. This is problematic in the context of control problems, where12

the control is determined by comparing value functions. In this paper we give a preprocessing13

step for Fourier methods which involves projecting the Green’s function onto the set of linear14

basis functions. The resulting algorithm is guaranteed to be monotone (to within a tolerance),15

`∞-stable and satisfies an ε-discrete comparison principle. In addition the algorithm has the16

same complexity per step as a standard Fourier method while at the same time having second17

order accuracy for smooth problems.18

Keywords: Monotonicity, Fourier methods, discrete comparison, optimal stochastic control,19

finance20

Running Title: ε-Monotone Fourier21

Key Messages:22

• Current Fourier methods (FST/CONV/COS) are not necessarily monotone23

• We devise a pre-processing step for FST/CONV methods which are monotone to a user24

specified tolerance25

• The resulting methods can be used safely for optimal control problems in finance26

1 Introduction27

Optimal stochastic control problems in finance often involve monitoring or making decisions at28

discrete points in time. These monitoring times typically cause difficulties when solving optimal29

∗David R. Cheriton School of Computer Science, University of Waterloo, Waterloo ON, Canada N2L 3G1,paforsyt@uwaterloo.ca, +1 519 888 4567 ext. 34415.†David R. Cheriton School of Computer Science, University of Waterloo, Waterloo ON, Canada N2L 3G1,

glabahn@uwaterloo.ca, +1 519 888 4567 ext. 34667

1

stochastic control problems numerically, both for efficiency and correctness. Efficiency because30

numerical methods are typically applied from one monitoring time to the next. Correctness arises31

as an issue when the decision is determined by comparing value functions, something problematic32

when discrete approximations are not monotone. These optimal stochastic problems arise in many33

important financial applications. This includes problems such as asset allocation (Li and Ng, 2000;34

Huang, 2010; Forsyth and Vetzal, 2017; Cong and Oosterlee, 2016), pricing of variable annuities35

(Bauer et al., 2008; Dai et al., 2008; Chen et al., 2008; Ignatieva et al., 2018; Alonso-Garcia et al.,36

2018; Huang et al., 2017), and hedging in discrete time (Remillard and Rubenthaler, 2013; Angelini37

and Herzel, 2014) to name just a few.38

These optimal control problems are typically modeled as the solutions of Partial Integro Differ-39

ential Equations (PIDEs), which can be solved via numerical, finite difference (Chen et al., 2008)40

or Monte Carlo (Cong and Oosterlee, 2016) methods. When cast into dynamic programming form,41

the optimal control problem reduces to solving a PIDE backwards in time between each decision42

point, and then determining the optimal control at each such point. In many cases, including for43

example those mentioned above, the models are based on fairly simple stochastic processes, with44

the main interest being the behaviour of the optimal controls. These simple stochastic models can45

be justified if one is looking at long term problems, for example, variable annuities or saving for46

retirement, where the time scales are of the order of 10-30 years. In these situations it is reasonable47

to use a parsimonious stochastic process model.48

In these (and many other) situations the characteristic function of the associated stochastic49

process is known in closed form. For the type of PIDEs that appear in financial problems, knowing50

the characteristic function implies that the Fourier transform of the solution is also known in closed51

form. By discretizing these Fourier transforms we obtain an approximation to the solution which52

can be used for effective numerical computation. A natural approach in this case is to use a Fourier53

scheme to advance the solution in a single time step between decision times, and then apply a54

numerical optimization approach to advance the solution across the decision time. This technique55

is repeated until the current time is reached (Ruijter et al., 2013; Lippa, 2013). These methods56

are based on Fourier Space Timestepping (FST) (Jackson et al., 2008), the CONV technique (Lord57

et al., 2008) or the COS algorithm (Fang and Oosterlee, 2008). Fourier methods have been applied58

to pricing of exotic variance products and volatility derivatives (Zheng and Kwok, 2014), guaranteed59

minimum withdrawal benefits (Ignatieva et al., 2018; Alonso-Garcia et al., 2018; Huang et al., 2017)60

and equity-indexed annuities (Deng et al., 2017) to name just a few.61

Fourier methods have a number of advantages compared to finite difference and other methods.62

First and foremost is that there are no timestepping errors between decision dates. These methods63

also provide easy handling for stochastic processes involving jump diffusion (Lippa (2013)) and64

regime switching (Jackson et al. (2008)). Although Fourier methods typically need a large number65

of discretization points, the algorithms reduce to using finite FFTs which are efficiently available66

on most platforms (including also GPUs). The algorithms are also quite easy to implement. For67

example, using Fourier methods for the pricing of variable annuities reduces to the use of discrete68

FFTs and local optimization. Detailed knowledge of PDE algorithms is not actually required in69

this case. Fourier methods also easily extend to multi-factor stochastic process where finite differ-70

ence methods have difficulties because of cross derivative terms. Of course, Fourier methods suffer71

from the curse of dimensionality, and hence are restricted, except in special cases, to problems of72

dimension three or less. Finally, Fourier methods have good convergence properties for problems73

with non-complex controls. For example, for European option pricing, in cases where the character-74

istic function of the underlying stochastic process is known, the COS method achieves exponential75

convergence (in terms of number of terms of the Fourier series) (Fang and Oosterlee, 2008).76

A major drawback with current, existing Fourier methods is that they are not monotone.77

2

In the contingent claims context, monotone methods preserve arbitrage inequalities, or discrete78

comparison properties, independent of any discretization errors. As a concrete example, consider79

the case of a variable annuity contract, with ratchet features and withdrawal controls at each80

decision date. Suppose contract A has a larger payoff at the terminal time than contract B. Then a81

monotone scheme generates a value for contract A which is always larger than the value of contract82

B, at all points in time and space, regardless of the accuracy of the numerical scheme. In a sense,83

the arbitrage inequality (discrete comparison) condition is the financial equivalence of conservation84

of mass in engineering computations. Use of non-monotone methods is especially problematic in85

the context of control problems, where the control is determined by comparing value functions.86

Monotonicity is also relevant for the convergence of numerical schemes. In general, optimal87

control problems posed as PIDEs are nonlinear and do not have unique solutions. The financially88

relevant solution is the viscosity solution of the PIDEs and it is well known (Barles and Souganidis,89

1991) that a discretization of a PIDE converges to the viscosity solution if it is monotone, consistent90

and stable. There are examples (Obermann, 2006) where non-monotone discretizations fail to91

converge and also examples where there is convergence (Pooley et al., 2003) but not to the financially92

correct viscosity solution. In addition, in cases where the Green’s function has a thin peak, existing93

non-monotonic Fourier methods require a very small space step which often results in numerical94

issues. Finally, monotone schemes are more reliable for the numeric computation of Greeks (i.e.95

derivatives of the solution), often important information for financial instruments.96

The starting point for this paper is the assumption that we have a closed form representation97

of the Fourier transform of the Green’s function of the stochastic process PIDE. From a practical98

point of view, we also assume that a spatial shift property also holds. This last assumption can99

be removed but at a cost of reducing the good computational complexity of our method. We will100

discuss these assumptions further in subsequent sections.101

In this paper we present a new Fourier algorithm in which monotonicity can be guaranteed to102

within a user specified numerical tolerance. The algorithm is for use with general optimal control103

problems in finance. In these general control problems the objective function may be complex104

and non-smooth, and hence the optimal control at each step must be determined by a numerical105

optimization procedure. Indeed, in many cases, this is done by discretizing the control and using106

exhaustive search. Reconstructing the Fourier coefficients is typically done by assuming the control107

is constant over discretized intervals of the physical space, numerically determining the control at108

the midpoint of these intervals and finally by reconstructing the Fourier coefficients by quadrature.109

This is equivalent to using a type of trapezoidal rule to reconstruct the Fourier coefficients and110

hence this can be at most second order accurate (in terms of the physical domain mesh size).111

In fact we show how one can modify the FST or CONV schemes to get new schemes in which112

monotonicity can be guaranteed to within a user specified numerical tolerance. Our approach is113

similar to that used in these schemes which first approximate the solution of a linear PIDE by a114

Green’s function convolution, then discretize the convolution and finally carry out the dense matrix-115

vector multiply efficiently using an FFT. In our case we discretize the value function, and generate116

a continuous approximation of the function by assuming linear basis (or alternatively piecewise117

constant) functions. Given this approximation, we carry out an exact integration of the convolution118

integral and then truncate the series approximation of this integral so that monotonicity holds to119

within a tolerance. Consequently, we prove that our algorithm has an ε−Discrete Comparison120

Property, that is, given a tolerance ε, then a discrete comparison (a.k.a. arbitrage inequality) holds121

to O(ε), independent of the number of discretization nodes and timesteps. This is similar in spirit122

to the ε-monotone schemes discussed in, for example, Bokanowski et al. (2018). Typically the123

convergence to the integral is exponential in the series truncation parameter so it is inexpensive124

to specify a small tolerance. The key idea here is that the number of terms required to accurately125

3

determine the projection of the Green’s function onto linear basis functions can be larger than126

the number of basis functions. After an initial set up cost, the complexity per step is the same127

as the standard FST or CONV methods. This requires only a small change to existing codes in128

order to guarantee monotonicity. The desirable property of our method is that monotonicity can129

be guaranteed (to within a tolerance) independent of the number of FST (CONV) grid nodes or130

time-step size.131

While Fourier methods have good convergence properties for vanilla contracts or problems where132

controls are smooth, it is a different story for general optimal control problems. For example, if the133

COS method is applied to optimal control problems, then it is challenging to maintain exponential134

convergence as the optimal control must be determined in the physical space. Hence, a highly135

accurate recursive expression for the Fourier coefficients must be found after application of the136

optimal controls, in order to maintain exponential convergence. In the case of bang-bang controls,137

it is often possible to separate the physical domain into regions where the control is constant. If138

these regions are determined to high accuracy, then an accurate algorithm for recursive generation139

of the Fourier coefficients can be developed (Ruijter et al. (2013)). However, even for the case of an140

American option, this requires careful analysis and implementation (Fang and Oosterlee (2008)).141

Our interest is in general problems, where the control may not be of the bang-bang type, and we142

expect that such good convergence properties will not hold. In addition, in the path dependent143

case, the problem is usually converted into Markovian form through additional state variables. The144

dynamics of these state variables are typically represented by a deterministic equation (between145

monitoring dates). At monitoring dates, the state variable may have non-smooth jumps (e.g. cash146

flows) and hence the standard approach would be to discretize this state variable, and then to147

interpolate the value function across the monitoring dates. If linear interpolation is used, this also148

implies that the solution is at most second order accurate at a monitoring date.149

While monotone schemes have good numerical properties they appear to inherently be low order150

methods. However, it would seem that in the most general case, it is difficult to develop high order151

schemes for control problems. For example, in the COS method, this problem can be traced to152

the difficulty of reconstructing the Fourier coefficients after numerically determining the optimal153

control at discrete points in the physical space. Consequently, in this article we focus on FST or154

CONV techniques, which use straightforward procedures to move between Fourier space and the155

physical space (and vice versa).156

We illustrate the behaviour of our algorithm by comparing various implementations of FST/CONV157

on some model option pricing examples, in particular European and Bermudan options. In addition,158

we demonstrate the use of the monotone scheme methods on a realistic asset allocation problem.159

Our main conclusion is that for problems with complex controls, where we can expect fairly low160

order convergence to the solution, a small change to standard FST or CONV methods can be made161

which guarantees monotonicity, at least to within a user specified tolerance. This does not alter162

the order of convergence in this case, hence we can ensure a monotone scheme with only a slightly163

increased set up cost. After the initialization, the complexity per step of the monotone method is164

the same as the standard FST/CONV algorithm.165

The remainder of this paper is as follows. In the next section we describe our optimal control166

problem in a general setting. Section 3 is used to describe existing Fourier methods which allows167

us to contrast with our new monotone Fourier method presented in Section 4. The monotone168

algorithm for solving optimal control problems is then given in Section 5 with properties of the169

algorithm and proofs appearing in the following section. Wrap-around is an important issue for170

Fourier methods, particularly in the case of our control problems. Our method of minimizing such171

error is described in Section 7. Section 8 presents two numerical examples used to stress test the172

monotone algorithm. This is followed by an application of our algorithm to the multiperiod mean173

4

variance optimal asset allocation problem, a general optimal control problem well suited to our174

monotone methods. The paper ends with a conclusion and topics for future research.175

2 General Control Formulation176

In this section we describe our optimal control problem in a general setting. Consider a set of177

intervention (or monitoring) times tn178

T ≡ {t0 ≤ · · · ≤ tM} (2.1)

with t0 = 0 the inception time of the investment and tM = T the terminal time. For simplicity, we179

specify the set of intervention times to be equidistant, that is, tn − tn−1 = ∆t = T/M for each n.180

Let t−n = tn − ε and t+n = tn + ε, with ε → 0+, denote the instant before and after the nth181

monitoring time tn. We define a value function v(x,t) with domain x ∈ R (we restrict attention to182

one dimensional problems for ease of exposition), which satisfies183

vt + Lv = 0 ; t ∈ (t+n , t−n+1) (2.2)

with L a partial integro-differential operator. At tn ∈ T we find an optimal control c(x,tn) via184

v(x, t−n ) = infc∈ZM(c)v(x, t+n ) (2.3)

where M(c) is an intervention operator and Z is the set of admissible controls.185

It is more natural to rewrite these equations going backwards in time τ = T − t, that is, in186

terms of time to completion. In this case the value function is v(x, τ) = v(x, T − t) and satisfies187

vτ − Lv = 0 ; τ ∈ (τ+n , τ

−n+1) (2.4)

v(x, τ+n ) = inf

cM(c)v(x, τ−n ) ; τn ∈ T . (2.5)

Here the control c(x, τ) = c(x, T − τ) and T now refers to the set of backwards intervention times188

T ≡ {τ0 ≤ · · · ≤ τM} with τ0 = 0, τM = T and τn = T − tM−n .

A typical intervention operator has the form189

M(c)v(x, τ−n ) = v(x+ Γ(x, τ−n , c), τ−n ) . (2.6)

As an example, in the context of portfolio allocation, we can interpret Γ(x, τ−i , c) as a rebalancing190

rule. In general, there can also be cash flows associated with the decision process, as in the case191

of variable annuities. However, for simplicity we will ignore such a generalization in this paper,192

and assume that the intervention operator has the form (2.6). In our asset allocation example193

(described later), the cash flows are modeled by updating a path dependent variable.194

3 CONV and FST Methods195

In this section, we derive the FST and closely related CONV technique in an intuitive fashion. This196

will allow us to contrast these methods with the monotone technique developed in the following197

section. For ease of exposition, we will continue to restrict attention to one dimensional problems.198

However, there is no difficulty generalizing this approach to the multi-dimensional case. In a199

financial context, we often have that the variable x = log(S) ∈ (−∞,∞), where S is an asset price.200

5

3.1 Green’s Functions201

A solution of the PIDE (2.4)202

vτ − Lv = 0 ; τ ∈ (τn, τn+1]

can be represented in terms of the Green’s function of the PIDE, a function typically of the form203

g = g(x, x′,∆τ). However, in many cases this Green’s function will have the form g = g(x−x′,∆τ)204

and we will assume this to hold in our problems. More formally, we make the following assumptions,205

which we assume to hold in the rest of this work.206

Assumption 3.1 (Form of Green’s function). We assume that the Green’s function can be written207

as208

g(x, x′,∆τ) = g(x− x′,∆τ)

=

∫ ∞−∞

G(ω,∆τ)e2πiω(x−x′) dω (3.1)

where G(ω,∆τ) is known in closed form, and G(ω,∆τ) is independent of (x, x′).209

Remark 3.1 (Assumption 3.1). If we view the Green’s function as a scaled conditional proba-210

bility density f then our assumption is that f(y|x) only depends on x and y via their difference211

f(y|x) = f(y − x). This assumption holds for Levy processes (independent and stationary in-212

crements), but does not hold, for example, for a Heston stochastic volatility model nor for mean213

reverting Ornstein-Uhlenbeck processes (but see Surkov (2010); Zhang et al. (2012); Shan (2014)214

for possible work-arounds). The ε-monotonicity modifications described in this paper also hold when215

we do not have g(x, x′,∆τ) = g(x− x′,∆τ) but at the price of reduced efficiency. This is discussed216

later in Section 4.2. The second assumption, that we know the Fourier transform of our Green’s217

function in closed form, is the case, for example, in situations where the characteristic function218

of the underlying stochastic process is known. In the case of Levy processes, the Levy-Khintchine219

formula provides such an explicit representation of the characteristic function.220

221

From Assumption 3.1, the exact solution of our PIDE is then222

v(x, τ + ∆τ) =

∫Rg(x− x′,∆τ)v(x′, τ) dx′ . (3.2)

The Green’s function has a number of important properties (Garroni and Menaldi, 1992). For this223

work the two properties224 ∫Rg(x,∆τ) dx = C1 ≤ 1 and g(x,∆τ) ≥ 0 (3.3)

are particularly important.1 These properties are formally proven in (Garroni and Menaldi, 1992),225

but can also be deduced from the interpretation of the Green’s function as a scaled probability226

density.227

We define the Fourier transform pair for the Green’s function as228

G(ω,∆τ) =

∫ ∞−∞

g(x,∆τ)e−2πiωx dx

g(x,∆τ) =

∫ ∞−∞

G(ω,∆τ)e2πiωx dω (3.4)

1For the examples in this paper the constant C1 is explicitly given (in each example) in Appendix A

6

with a closed form expression for G(ω,∆τ) being available.229

As is typically the case, we assume that the Green’s function g(x,∆τ) decays to zero as |x| → ∞,230

that is, g(x,∆τ) is negligible outside a region x ∈ [−A,A]. Choosing xmin < −A and xmax > A, we231

localize the computational domain for the integral in equation (3.2) so that x ∈ [xmin, xmax]. We232

can therefore replace the Fourier transform pair (3.4) by their Fourier series equivalent233

G(ωk,∆τ) '∫ xmax

xmin

g(x,∆τ) e−2πiωkx dx

g(x,∆τ) =1

P

∞∑k=−∞

G(ωk,∆τ) e2πiωkx (3.5)

with P = xmax − xmin and ωk = kP . Here the scaling factors in equation (3.5) are selected to be234

consistent with the scaling in (3.4). The solution of the PIDE (3.2) is then approximated as235

v(x, τ + ∆τ) '∫ xmax

xmin

g(x− x′,∆τ)v(x′, τ) dx′. (3.6)

Note that the Fourier series (3.5) implies a periodic extension of g, that is, g(x + P, τ) = g(x, τ).236

The localization assumption also then implies that v(x, τ) is periodically extended.237

Substituting the Fourier series (3.5) into (3.6) gives our approximate solution as238

v(x, τ + ∆τ) ' 1

P

∞∑k=−∞

G(ωk,∆τ)e2πiωkx

∫ xmax

xmin

v(x′, τ)e−2πiωkx′dx′. (3.7)

Let ∆x = PN and choose points {xj}, {x′j} by239

xj = x0 + j∆x ; x′j = x0 + j∆x for j = −N/2, . . . N/2− 1.

Then the integral in (3.7) can be approximated by a quadrature rule with weights w` giving240 ∫ xmax

xmin

v(x′, τ)e−2πiωkx′dx′ '

N/2−1∑`=−N/2

w` v(x′`, τ)e−2πi kPx′`∆x

= Pe−2πi kPx0V (ωk, τ) (3.8)

where241

V (ωk, τ) =1

N

N/2−1∑`=−N/2

w` v(x′`, τ)e−2πik`/N (3.9)

is the DFT of {wj v(x′j , τ)}. In the following, we will consider two cases for the weights w`: the242

trapezoidal rule and Simpson’s quadrature. Using equations (3.8) and (3.9) in equation (3.7), and243

truncating the infinite sum to k ∈ [−N/2, N/2− 1] then gives244

v(xj , τ + ∆τ) ' 1

P

N/2−1∑k=−N/2

e2πi kPx0G(ωk,∆τ)e2πikj/NPe−2πi k

Px0V (ωk, τ)

=

N/2−1∑k=−N/2

G(ωk,∆τ)V (ωk, τ)e2πikj/N . (3.10)

7

Thus {v(xj , τ + ∆τ)} is the inverse DFT of the product {G(ωk,∆τ) · V (ωk, τ)}.245

In summary, one can obtain a discrete set of values for the solution v by first going to the246

Fourier domain by constructing its Fourier transform V using a set of quadrature weights and then247

returning to the physical domain by convolution of V with the Fourier transform of the Green’s248

function. The cost is then the cost of doing a single FFT and iFFT.249

There are four significant approximations in these steps. These include localization of the250

computational domain, representation of the Green’s function by a truncated Fourier series, a251

periodic extension of the solution and, finally, approximation of the integral in equation (3.7) by a252

quadrature rule. The effect of the errors from these approximations has been previously discussed253

in Lord et al. (2008) and we refer the reader there for details.254

3.2 The FST/CONV Algorithms255

The FST and CONV algorithms are described using the previous approximations. Let (vn)+ be256

the vector of solution values just after τn and wquad be the vector of quadrature weights257

(vn)+ = [v(x−N/2, τ+n ), . . . , v(xN/2−1, τ

+n )]

wquad = [w(x−N/2), . . . , w(xN/2−1)] . (3.11)

Furthermore let I∆x(x), with xk ≤ x ≤ xk+1, be a linear interpolation operator258

I∆x(x)(vn)+ = θ · v(xk, τ+n ) + (1− θ) · v(xk+1, τ

+n ) with θ =

(xk+1 − x)

∆x. (3.12)

The full FST/CONV algorithm applied to a control problem is illustrated in Algorithm 1. We refer259

the reader to Lippa (2013); Ignatieva et al. (2018); Huang et al. (2017) for applications in finance.260

Algorithm 1 FST/CONV Fourier method. x ◦ y is the Hadamard product of vectors x,y.

Require: G = {G(ωj ,∆τ)}, j = −N/2, . . . , N/2− 11: Input: number of timesteps M and initial solution (v0)−

2: (v0)+ = infcM(c)(I∆x(x)(v0)−

)3: for m = 1, . . . ,M do { Timestep loop}4: V m−1 = FFT [ wquad ◦ (vm−1)+ ] {Frequency domain}5: (vm)− = iFFT [ V m−1 ◦G ] {Physical domain}6: v(xj , τ

+m) = infcM(c)

(I∆x(xj)(v

m)−)

; j = −N2 , . . . , j = N

2 − 1 {Optimal control}7: end for

Remark 3.2. In (Jackson et al., 2008), the authors describe their FST method in slightly different261

terms. There they use a continuous Fourier transform to convert the PIDE into Fourier space.262

The PIDE in physical space then reduces to a linear first-order differential equation in Fourier263

space which can be solved in closed-form as long as the characteristic function of the associated264

stochastic process is known in closed form (See Appendix A). In this way, the method is able to265

produce exact pricing results between monitoring dates (if any) of an option, using a continuous266

domain. In practice, using a discrete computational domain leads to approximations as a discrete267

Fourier transform is used to approximate the continuous Fourier transform.268

4 An ε-Monotone Fourier Method269

Our monotone Fourier method proceeds in a similar fashion as in the previous section, but is based270

on a slightly different philosophy. We begin by discretizing the value function, and then generate271

8

a continuous approximation of the value function by assuming linear basis functions. Given this272

approximation, we carry out an exact integration of the convolution integral. We can then truncate273

the series approximation of this integral so that monotonicity holds to within a tolerance (using274

a truncation parameter to keep track of the number of terms). Typically the convergence to the275

integral is exponential in the series truncation parameter, so it is inexpensive to specify a small276

tolerance. The key idea here is that the number of terms required to accurately determine the277

projection of the Green’s function onto a given set of linear basis functions can be larger than the278

number of basis functions.279

An additional important point is that, after the initial set-up cost, the complexity per step is280

the same as the standard FST or CONV methods. This requires only a small change to existing281

FST or CONV codes in order to guarantee monotonicity. The desirable property of this method is282

that monotonicity can be guaranteed (to within a small tolerance) independent of the number of283

FST (CONV) grid nodes or timestep size.284

4.1 A Monotone Scheme285

We proceed as follows. As before we assume a localized computational domain286

v(x, τ + ∆τ) =

∫ xmax

xmin

g(x− x′,∆τ)v(x′, τ) dx′ (4.1)

and discretize this problem on the grid {xj}, {x′j}287

xj = x0 + j∆x ; x′j = x0 + j∆x ; j = −N2, . . . ,

N

2− 1

where P = xmax − xmin and ∆x = PN with xmin = x0 − N∆x

2 and xmax = x0 + N∆x2 . Setting288

vj(τ) = v(xj , τ) we can now represent the solution as a linear combination289

v(x, τ) 'N/2−1∑j=−N/2

φj(x) v(xj , τ) =

N/2−1∑j=−N/2

φj(x) vj(τ) . (4.2)

where the φj are piecewise linear basis functions, that is,290

φj(x) =

(xj+1−x)

∆x xj ≤ x ≤ xj+1(x−xj−1)

∆x xj−1 ≤ x ≤ xj0 otherwise .

(4.3)

Substituting representation (4.2) into equation (4.1) gives291

vk(τ + ∆τ) =

∫ xmax

xmin

g(xk − x,∆τ) v(x, τ) dx

=

N/2−1∑j=−N/2

vj(τ)

∫ xmax

xmin

φj(x) g(xk − x,∆τ) dx

=

N/2−1∑j=−N/2

vj(τ) g(xk − xj ,∆τ)∆x , (4.4)

9

where292

g(xk − xj ,∆τ) =1

∆x

∫ xk−xj+∆x

xk−xj−∆xφk−j(x) g(x,∆τ) dx . (4.5)

Here we have used the fact that φj(xk−x) = φk−j(x), a property which follows from the properties293

of linear basis functions. Setting ` = k − j, y` = xk − xj = `∆x for ` = −N2 , . . . ,

N2 − 1 gives294

g(y`,∆τ) =1

∆x

∫ y`+∆x

y`−∆xφ`(x) g(x,∆τ) dx (4.6)

as the averaged projection of the Green’s function onto the basis functions φ`. Note that for this295

projection g(y`,∆τ) ≥ 0 since the exact Green’s function has g(x) ≥ 0 for all x, and of course296

φ`(x) ≥ 0. Therefore the scheme (4.4) is monotone for any N .297

298

Remark 4.1 (Green’s function available in closed form). If the Green’s function is available in299

closed form, rather than just its Fourier Transform, then equation (4.6) can be used to directly300

compute the g(y`,∆τ) terms, as, for example, in Tanskanen and Lukkarinen (2003). However, in301

general this will require a numerical integration. If the Fourier transform of the Green’s function is302

known, we will derive a technique to efficiently compute g(y`,∆τ) to an arbitrary level of accuracy.303

304

4.2 Approximating the Monotone Scheme305

The scheme (4.4) is monotone since the weights g(y`,∆τ) given in (4.6) are nonnegative. However306

it is only possible for us to approximate these weights and this prevents us from guaranteeing307

monotonicity. In this subsection we show how we overcome this issue.308

Recall that our starting point is that G, the Fourier series of the Green’s function, is known309

in closed form. We have then replaced our Green’s function g(x,∆τ) by its localized, periodic310

approximation311

g(x,∆τ) =1

P

∞∑k=−∞

e2πiωkxG(ωk,∆τ) where ωk =k

Pand P = xmax − xmin

and then projected the Green’s function onto the linear basis functions. Replacing g(x,∆τ) by312

g(x,∆τ) in equation (4.6), and assuming uniform convergence of the Fourier series (see Appendix),313

we integrate equation (4.6) term by term resulting in314

g(yj ,∆τ) =1

P

∞∑k=−∞

(1

∆x

∫ yj+∆x

yj−∆xe2πiωkxφj(x) dx

)G(ωk,∆τ). (4.7)

In the case of linear basis functions (4.3) we convert the complex exponential in (4.7) into trigono-315

metric functions with the resulting integration giving2316

g(yj ,∆τ) =1

P

∞∑k=−∞

e2πiωkyj

(sin2 πωk∆x

(πωk∆x)2

)G(ωk,∆τ). (4.8)

2For ωk = 0, we take the limit ωk → 0.

10

This is then approximated by truncating the series.317

A key point is that the truncation of the projection of the Green’s function does not have to318

use the same number of terms as the number of basis functions. That is, set N ′ = αN , with N319

defined in equation (4.2) and α = 2k for k = 1, 2, . . .. Suppose we now truncate the Fourier series320

for the projected linear basis form for g to N ′ terms. Let g(yk,∆τ, α), denote the use of a truncated321

Fourier series with truncation parameter α for a fixed value of N so that the Fourier series (4.8)322

truncates to323

g(yj ,∆τ, α) =1

P

αN/2−1∑k=−αN/2

e2πiωkj∆x

(sin2 πωk∆x

(πωk∆x)2

)G(ωk,∆τ) . (4.9)

Using the notation gj(∆τ, α) = g(yj ,∆τ, α), then324

gj+N (∆τ, α) = gj(∆τ, α)

so that our sequence {g−N/2(∆τ, α), . . . , gN/2−1(∆τ, α)} is periodic.325

326

Remark 4.2 (Efficient computation of the projections). It remains to compute the projections.327

For this one needs to determine the discrete convolution (4.9). Let328

Yk =

(sin2 πωk∆x

(πωk∆x)2

)G(ωk,∆τ) ; k = −αN

2, . . . ,

αN

2− 1.

Then rewriting e2πiωkj∆x = e2πik`/(αN) with ` = jα and defining329

Y` =1

P

αN/2−1∑k=−αN/2

e2πik`/(αN)Yk ; ` = −αN2, . . . ,

αN

2− 1 (4.10)

gives {Y`} as the DFT of the {Yk} (of size N ′ = αN). Consequently, using equations (4.9) and330

(4.10)331

g(yj ,∆τ, α) = Y` ; ` = jα ; j = −N/2, . . . , N/2− 1 . (4.11)

Thus the projections {g(yj ,∆τ, α)} are computed via a single FFT of size N ′.332

333

For k = −N2 , . . . ,

N2 − 1 we define G(ωk,∆τ, α) as the DFT of the {g(ym,∆τ, α)}334

G(ωk,∆τ, α) =P

N

N/2−1∑m=−N/2

e−2πimk/N g(ym,∆τ, α). (4.12)

Note that335

∆x =P

N>

P

N ′=

P

αN

that is, the basis function is integrated over a grid of size ∆x > P/N ′, and so is larger than the grid336

spacing on the N ′ grid. As α→∞, there is no error in evaluating these integrals (projections) for337

a fixed value of N . For any finite α, there is an error due to the use of a truncated Fourier Series.338

11

Again, we emphasize that the truncation for the Fourier series representation of the projection339

of the Green’s function in (4.9) does not have to use the same number of terms (αN) as used in the340

discrete convolution (N). Instead we can take a very accurate expansion of the Green’s function341

projection and then translate this back to the coarse grid using (4.12). There is no further loss342

of information in this last step. As remarked above, we only use the Fourier representation of343

g(yj ,∆τ, α) to carry out the discrete convolution, that is a dense matrix vector multiply, efficiently.344

The discrete convolution in Fourier space is exactly equivalent to the discrete convolution in physical345

space, assuming periodic extensions.346

347

Remark 4.3 (Assumption 3.1 revisited). The assumption that g(x, x′,∆τ) = g(x − x′,∆τ) will348

permit fast computation of a dense matrix-vector multiply using an FFT. As mentioned earlier349

this assumption holds for Levy processes, but does not hold, for example, for a Heston stochastic350

volatility model. However the basic idea of projection of the Green’s function onto linear basis351

functions can be used even if Assumption 3.1 does not hold. The price, in this case, is a loss352

of computational efficiency. As an example, in the case of the Heston stochastic volatility model,353

one has a closed form for the characteristic function but here the Green’s function has the form354

g = g(ν, ν ′, x − x′,∆τ) where ν is the variance and x = logS, where S is the asset price. In this355

case, we can use an FFT effectively in the x direction, but not in the ν direction.356

357

Remark 4.4 (Relation to the COS method). In the COS method, the solution v(x,τ) is also ex-358

panded in a Fourier series. This gives exponential convergence of the entire algorithm for smooth359

v(x,τ) which in turn requires that we have a highly accurate Fourier representation of v(x,τ). How-360

ever, suppose v(x,τ) is obtained from applying an impulse control using a numerical optimization361

method at discrete points on a previous step, using linear interpolation (the only interpolation362

method which is monotone in general). In that case we will not have an accurate representation of363

the Fourier series of v(x,τ). In addition, it does not seem possible to ensure monotonicity for the364

COS method. So far, we have only assumed that the v(x, τ) can be expanded in terms of piecewise365

linear basis functions. This property can be used to guarantee monotonicity. However convergence366

will be slower than the COS method if the solution is smooth.367

Remark 4.5 (Piecewise constant basis functions). The equations and previous discussion in this368

section also holds if our basis functions are piecewise constant functions, that is, basis functions φj369

which are nonzero over [xj −∆x/2, xj + ∆x/2]. In this case computing the integral in (4.7) gives370

g(yj ,∆τ) =1

P

∞∑k=−∞

e2πiωkyj

(sinπωk∆x

πωk∆x

)G(ωk,∆τ) (4.13)

with the subsequent equations also requiring slight modifications.371

12

4.3 Computing the Monotone Scheme372

In order to ensure our monotone approach is effective, it remains to compute the discrete convolution373

(4.4) efficiently. For the DFT pair for vj(τ) and V (ωp, τ), we recall that xj = x0 + j∆x and so374

vj(τ) =

N/2−1∑`=−N/2

V (ω`, τ)e2πiωlxj =

N/2−1∑`=−N/2

(e2πiω`x0

)V (ω`, τ)e2πij`/N

V (ωp, τ) =1

N

N/2−1∑`=−N/2

e−2πiωpx`v`(τ) =1

N

(e−2πiωpx0

) N/2−1∑`=−N/2

e−2πip`/Nv`(τ) . (4.14)

Suppose we write g(xk − xj ,∆τ) as a DFT375

gk−j(∆τ, α) =1

P

N/2−1∑p=−N/2

G(ωp,∆τ, α)e2πi(k−j)p/N , (4.15)

where we use equation (4.12) to determine G(ωp,∆τ, α). Substituting equations (4.15) and (4.14)376

into equation (4.4) we then get377

v(xk, τ + ∆τ) = ∆x

N/2−1∑j=−N/2

vnj g(xk − xj ,∆τ, α)

=1

N

N/2−1∑p=−N/2

N/2−1∑`=−N/2

(e2πiω`x0

)G(ωp,∆τ, α)V (ω`, τ)e2πikp/N

N/2−1∑j=−N/2

e2πij(`−p)/N

=

N/2−1∑p=−N/2

(e2πiωpx0

)V (ωp, τ)G(ωp,∆τ, α)e2πikp/N (4.16)

where the last equation follows from the classical orthogonality properties of N th roots of unity.378

From equation (4.14) we have379

V (ωp, τ) =1

N

(e−2πiωpx0

) N/2−1∑`=−N/2

e−2πip`/Nv`(τ) =

(e−2πiωpx0

)V (ωp, τ) (4.17)

with380

V (ωp, τ) =1

N

N/2−1∑`=−N/2

e−2πip`/Nv`(τ)

the DFT of {v`(τ)}. Finally substituting equation (4.17) into (4.16) gives381

v(xk, τ + ∆τ) =

N/2−1∑p=−N/2

V (ωp, τ) G(ωp,∆τ, α) e2πip`/N , (4.18)

which we recognize as the inverse DFT of {V (ωp, τ) G(ωp,∆τ, α)}.382

13

Remark 4.6 (Monotonicity). Equations (4.18) and (4.4) are algebraic identities (assuming peri-383

odic extensions). Hence if we use (4.18) to advance the solution, then this is algebraically identical384

to using (4.4) to advance the solution. Thus we can analyze the properties of equation (4.18) by385

analyzing equation (4.4). In particular, if g(xk,∆τ, α) ≥ 0 then the scheme is monotone.386

Remark 4.7 (Converting FST or CONV to monotone form). Equation (4.18) is formally identical387

with equation (3.10). This has the practical result that any FST or CONV software can be converted388

to monotone form by a preprocessing step which computes G(ωp,∆τ, α), and choosing a trapezoidal389

rule for the integral in equation (3.8).390

5 Monotone algorithm for solution of the control problem391

In this section we describe our monotone algorithm for the control problem (2.4 - 2.5). Let (vn)+392

be the vector of values of our solution just after τn as defined earlier in equation (3.11) and I∆x(x)393

the linear interpolation operator defined as in equation (3.12). Let394

V n = [V (ω−N/2, τn), . . . , V (ωN/2−1, τn)] = DFT [ (vn)− ]

and395

G = [G(ω−N/2,∆τ, α), . . . , G(ωN/2−1,∆τ, α)] .

Let us assume that our Green’s function is not an explicit function of τ but rather we have396

g = g(x − x′,∆τ) and that the time steps are all constant, that is, τn+1 − τn = ∆τ = const.397

In this case we can compute G(ωk,∆τ, α) only once. If these two assumptions do not hold, then398

G(·) would have to be recomputed frequently and hence our algorithm for ensuring monotonicity399

becomes more costly.400

Algorithm 2 describes the computation of G(·). Here we test for monotonicity (up to a smalltolerance) by minimizing the effect of any negative weights which are determined via∑

j

∆x|min(g(yj ,∆τ, α), 0)| < ε1∆τ

T.

The test for accuracy of the projection occurs by the comparison

maxj

∆x|g(yj ,∆τ, α)− g(yj ,∆τ, α/2)| < ε2.

Both monotonicity and convergence tests are scaled by ∆x so that these quantities are bounded as401

∆τ → 0,∀∆x (the Green’s function becomes unbounded as ∆τ → 0, but the integral of the Green’s402

function is bounded by unity). In addition, the monotonicity test scales ε1 by ∆τ/T in order to403

eliminate the number of timesteps from our monotonicity bounds. This is also discussed further in404

Section 6.405

In Algorithm 2, the test on line 5 will ensure that monotonicity holds to a user specified tolerance406

and the test on line 6 ensures accuracy of the projections. The complete monotone algorithm for407

the control problem is given in Algorithm 3.408

Remark 5.1 (Convergence of Algorithm 2.). In Appendix B we show that for typical Green’s409

functions, the test for monotonicity on line 5 in Algorithm 2 and the accuracy test on line 6 are410

usually satisfied for α = 2, 4, for typical values of ε1, ε2.411

Remark 5.2 (Complexity). The complexity of using (4.18) to advance the time (excluding the412

cost of determining an optimal control) is O(N logN) operations, roughly the same as the usual413

FST/CONV methods.414

14

Algorithm 2 Initialization of the monotone Fourier method.

Require: Closed form expression for G(ω,∆τ), the Fourier transform of the Green’s Function1: Input: N,∆x,∆τ2: Let α = 1 and compute g(yj ,∆τ, 1) .3: for α = 2k; k = 1,2, . . . until convergence do {Construct accurate g}4: Compute g(yj ,∆τ, α), G(ωj ,∆τ, α), j = −N

2 , . . . ,N2 − 1 using (4.11)-(4.12)

5: test1 =∑j

∆xmin(g(yj ,∆τ, α), 0) { Monotonicity test }

6: test2 = maxj

∆x|g(yj ,∆τ, α)− g(yj ,∆τ, α/2)| { Accuracy test }

7: if (|test1| < ε1(∆τ/T )) and (test2 < ε2) then8: break from for loop { Convergence test }9: end if

10: end for{End accurate g loop}11: Output: Weights G(ωj ,∆τ, α), j = −N

2 , . . . ,N2 − 1 in Fourier domain.

Algorithm 3 Monotone Fourier method.

Require: Weights G = {G(ωj ,∆τ, α)}, for j = −N2 , . . . ,

N2 −1 in Fourier domain (from Algor. 2).

1: Input: number of timesteps M , initial solution (v0)−

2: (v0)+ = infcM(c)(I∆x(x)(v0)−

)3: for m = 1, . . . ,M do {Timestep loop}4: V m−1 = FFT [ (vm−1)+ ] {Frequency domain}5: (vm)− = iFFT [ V m−1 ◦ G ] {Physical domain}6: v(xj , τ

+m) = infcM(c)

(I∆x(xj)(v

m)−)

; j = −N2 , . . . ,

N2 − 1 {Optimal control}

7: end for{End timestep loop}

6 Properties of the Monotone Fourier Method415

In this section we prove a number of properties satisfied by our ε-monotone Fourer algorithm. The416

main properties include `∞ stability and a type of ε-discrete comparison principle.417

Lemma 6.1. Let C1 be a constant such that the exact Green’s function satisfies C1 =∫R g(x,∆τ) dx.418

Then for all k419

∆x

N/2−1∑j=−N/2

g(xk − xj ,∆x, α) = C1 with ∆x =P

N.

15

Proof.

∆x

N/2−1∑j=−N/2

g(xk − xj ,∆x, α) =P

N

N/2−1∑`=−N/2

g(y`,∆τ, α) ; where y` = xk − xj , ` = k − j

=P

N

N/2−1∑`=−N/2

1

P

αN/2−1∑k=−αN/2

e2πiωk`∆x

(sin2 πωk∆x

(πωk∆x)2

)G(ωk,∆τ)

=1

N

αN/2−1∑k=−αN/2

(sin2 πωk∆x

(πωk∆x)2

)G(ωk,∆τ)

N/2−1∑`=−N/2

e2πi`k/N

= G(0,∆τ) =

∫ ∞−∞

g(x,∆τ) dx = C1 .

420

Theorem 6.1 (`∞ stability). Assume that G is computed using Algorithm 2, that (vn)− is computed421

from422

(vnk )− =

N/2−1∑j=−N/2

∆x gk−j(vn−1j )+ , (6.1)

and that423

‖(vn)+‖∞ ≤ ‖(vn)−‖∞ . (6.2)

Then for every 0 ≤ n ≤M we have424

‖(vn)+‖∞ ≤ C2 = e2ε1‖(v0)−‖∞ .

Proof. From equation (6.1)425

(vnk )− =

N/2−1∑j=−N/2

∆x gk−j(vn−1j )+

=

N/2−1∑j=−N/2

∆x max(gk−j ,0)(vn−1j )+ +

N/2−1∑j=−N/2

∆x min(gk−j ,0)(vn−1j )+ . (6.3)

which then implies426

|(vnk )−| ≤ ‖(vn−1)+‖∞N/2−1∑j=−N/2

∆x max(gk−j , 0) + ‖(vn−1)+‖∞N/2−1∑j=−N/2

∆x |min(gk−j ,0)| .

From Lemma 6.1427

N/2−1∑j=−N/2

∆x max(gk−j , 0) = C1 +

N/2−1∑j=−N/2

∆x |min(gk−j , 0)| (6.4)

16

and so428

|(vnk )−| ≤ ‖(vn−1)+‖∞(C1 + 2

N/2−1∑j=−N/2

∆x |min(gk−j , 0)|)

≤ ‖(vn−1)+‖∞(C1 + 2ε1∆τ

T) . (6.5)

Using lines 5 and 7 in Algorithm 2. Since equation (6.5) is true for any k we have that429

‖(vn)−‖∞ ≤ ‖(vn−1)+‖∞(C1 + 2ε1∆τ

T) ,

which combined with equation (6.2) and using C1 ≤ 1 gives430

‖(vn)+‖∞ ≤ ‖(vn−1)+‖∞(1 + 2ε1∆τ

T) .

Iterating the above bound and using equation (6.2) at n = 0 gives431

‖(vn)+‖∞ ≤ ‖(v0)−‖∞(1 + 2ε1∆τ

T)n

≤ ‖(v0)−‖∞ e2ε1n∆τT ≤ ‖(v0)−‖∞ e2ε1 = C2 .

432

Remark 6.1 (Jump condition). We remark that (6.2), the jump condition ‖(vn)+‖∞ ≤ ‖(vn)−‖∞,433

is trivially satisfied if I∆x(x) in line 6 in Algorithm 3 is a linear interpolant.434

From Theorem 6.1 and Remark 6.1 we have the immediate result,435

Corollary 6.1 (Stability of Algorithm 3). Algorithm 3 is `∞ stable.436

Lemma 6.2 (Minimum value of solution.). Let (vn)+ be generated using equation 6.1, and set437

(vn)+min = min

k(vnk )+ .

If the conditions for Lemma 6.1 are satisfied and438

(vn)+min ≥ (vn)−min , (6.6)

then439

(vn)+min ≥ (v0)−min (C3)n − C2(eε1 − 1)

where C2 = ‖(v0)−‖∞e2ε1 is given in Lemma 6.1 and C3 =∑N/2−1

j=−N/2 ∆x max(gk−j ,0).440

Proof. From equation (6.1) and using equation (6.3) along with the definition of C3 we obtain441

(vnk )− ≥ (vn−1)+min

N/2−1∑j=−N/2

∆x max(gk−j ,0) +

N/2−1∑j=−N/2

∆x min(gk−j ,0)(vnj )+

≥ (vn−1)+min

N/2−1∑j=−N/2

∆x max(gk−j ,0)− ‖(vn−1)+‖∞N/2−1∑j=−N/2

∆x |min(gk−j ,0)|

= (vn−1)+minC3 − ‖(vn−1)+‖∞

( N/2−1∑j=−N/2

∆x |min(gk−j , 0)|).

17

Using Lemma 6.1 and lines 5 and 7 in Algorithm 2 then gives442

(vnk )− ≥ (vn−1)+minC3 − C2ε1

∆τ

T

and, since this is valid for any k, using (6.6) we obtain443

(vn)+min ≥ (vn−1)+

minC3 − C2ε1∆τ

T.

Iterating implies444

(vn)+min ≥ (v0)+

minC3n − C2ε1

∆τ

T

(1− C3

n

1− C3

)≥ (v0)−minC3

n − C2ε1∆τ

T

(1− C3

n

1− C3

), (6.7)

where we again use equation (6.6) in the last line. From equation (6.4) and the definition of C3 we445

have446

C3 = C1 +

N/2−1∑j=−N/2

∆x |min(gk−j , 0)| ≤ 1 + ε1∆τ

T, (6.8)

where the last inequality follows lines 5 and 7 in Algorithm 2 (and recalling that C1 ≤ 1). Combining447

equations (6.6), (6.7) and (6.8 ) and noting that n∆τ ≤ T gives448

(vn)+min ≥ (v0)+

minC3n − C2(eε1 − 1) ≥ (v0)−minC3

n − C2(eε1 − 1) .

449

Remark 6.2. We note that condition 6.6, that is, (vn)+min ≥ (vn)−min is satisfied if I∆x(x) in line 6450

in Algorithm 3 is a linear interpolant.451

Theorem 6.2 (ε-Discrete Comparison Principle). Suppose we have two independent discrete solu-452

tions453

(un)+ = [u(x−N/2, τ+n ), . . . , u(x+N/2−1, τ

+n )]

(wn)+ = [w(x−N/2, τ+n ), . . . , w(x+N/2−1, τ

+n )] (6.9)

with454

(u0)− ≥ (w0)− (6.10)

where the inequality is understood in the component-wise sense, and (un)+, (wn)+ are computed455

using Algorithm 3. If G is computed using Algorithm 2 and I∆x(x) is a linear interpolant then456

(un)+ − (wn)+ ≥ −ε1‖(u0 − w0)−‖∞ +O(ε21) ; ε1 → 0 . (6.11)

18

Proof. Let (zn)+ = (un)+ − (wn)+, (zn)− = (un)− − (wn)−, then457

(znk )− =

N/2−1∑j=−N/2

∆x gk−j(zn−1j )+ .

Noting that458

zj(τ+n ) = inf

cM(c)

(I∆x(xj)(u

m)−)− inf

cM(c)

(I∆x(xj)(w

m)−)

(6.12)

then459

|zj(τ+n )| ≤ sup

cM(c)

∣∣ I∆x (xj)((um)− − (wm)−

) ∣∣ (6.13)

hence, using the definition of the intervention operator (2.6), we obtain460

‖(zn)+‖∞ ≤ ‖(zn)−‖∞ . (6.14)

Similarly461

(zn)+min = min

jzj(τ

+n )

≥ minj

infcM(c) I∆x(xj)

((um)− − (wm)−

)≥ (zn)−min . (6.15)

Hence condition (6.2) of Lemma 6.1 and condition (6.6) of Lemma 6.2 are satisfied. Applying462

Lemma 6.2 to (zn)+, (zn)− we get463

(zn)+min ≥ (z0)−min(C3)n − e2ε1‖(u0 − w0)−‖∞(eε1 − 1) (6.16)

where C3 =∑N/2−1

j=−N/2 ∆x max(gk−j ,0). Since (z0)−min ≥ 0 and 0 ≤ Cn3 ≤ eε1 , the result follows.464

Remark 6.3. If Algorithm 2 is used to construct G for use in Algorithm 3, then the ε-discrete465

comparison property is satisfied for any N,∆τ,M up to order ε1. Since typically g(yj ,∆τ, α) →466

g(yj ,∆τ,∞) ≥ 0 exponentially in α, in practice it is very inexpensive to make ε1 as small as desired.467

Remark 6.4 (Continuously observed impulse control problems). By determining the optimal con-468

trol at each timestep, we can apply our monotone Fourier method to the continuously observed469

impulse control problem470

max

[vτ − Lv, v − inf

cM(c)v

]= 0 . (6.17)

This is effectively a method whereby the optimal control is applied explicitly, as in (Chen and471

Forsyth, 2008). Using the methods developed in this paper combined with those from (Chen and472

Forsyth, 2008), it is straightforward to show that the ε-monotone Fourier technique is `∞ stable and473

consistent in the viscosity sense as ∆τ,∆x→ 0. The ε-monotone Fourier method is also monotone474

to O(h) where h = O(∆x) = O(∆τ) is the discretization parameter. Thus it is possible to show475

convergence to the viscosity solution using the results in Barles and Souganidis (1991) extended as476

in Azimzadeh et al. (2017), using the ε monotonicity property as in Bokanowski et al. (2018).477

19

7 Minimization of wrap-around error478

The use of the convolution form for our solution (4.18) is rigorously correct for a periodic extension479

of the solution and the Green’s function. In normal option pricing applications, the wrap-around480

error due to periodic extension causes little error. However, in control applications, the values used481

in the optimization step (2.5) may be near the ends of the grid and hence large errors may result482

(Lippa, 2013; Ruijter et al., 2013; Ignatieva et al., 2018). Hence we need to consider methods to483

reduce errors associated with wrap-around.484

In order to minimize the effect of wrap-around we proceed in the following manner. Given the485

localized problem on [xmin, xmax] with N nodes, we construct an auxiliary grid with Na = 2N486

nodes, on the domain [xamin, xamax] where487

xamin = xmin −(xmax − xmin)

2and xamax = xmax +

(xmax − xmin)

2(7.1)

with (xamax − xamin) = 2(xmax − xmin). We construct and store the DFT of the projection of the488

Green’s function G(ωp,∆τ, α), p = −Na/2, . . . , p = Na/2−1 on this auxiliary grid. We then replace489

line 4 in Algorithm 3 by applying the DFT to the solution v on the auxiliary grid490

v(xk, τ+n )a = v(xk, τ

+n ) ; k = −N/2, . . . , k = N/2− 1

= v(x−N/2, τ+n ) ; k = −Na/2, . . . ,−N/2− 1 (7.2)

= A(xk, τ+n ) ; k = N/2, . . . , Na/2− 1 , (7.3)

where A(x, τ) is an asymptotic form of the solution, which we assume to be available from financial491

reasoning. On the auxiliary grid near x → −∞ we simply extend the solution by the constant492

value at x = xmin, which is expected to generate a small error, since the grid spacing (in terms493

of S = ex) is very small. We then carry out lines 4 - 5 of Algorithm 3 on the auxiliary grid and494

generate (vn)− by discarding all the values on the auxiliary grid which are not on the original grid495

(as these are contaminated by wrap-around errors). The errors incurred by using extensions (7.2)496

and (7.3) can be made small by choosing |xmin| and xmax sufficiently large.497

Remark 7.1 (Use of asymptotic form to reduce wrap-around error). Use of the above technique498

necessitates some changes to the proof of Theorem 6.2. However, the main result is the same, with499

adjustments to some of the constants in the bounds. This is a tedious algebraic exercise which we500

omit.501

502

Remark 7.2 (Additional complexity to reduce wrap-around). For a one dimensional problem, the503

complexity for one timestep is O(Na logNa) = O(2N log(2N), where N is the number of nodes in504

the original grid. In the case of the path dependent problem in Section 9, if there are Nx nodes505

in the logS direction and Nb nodes in the bond direction, then the complexity for one timestep is506

O(2NbNx log(2Nx)).507

508

8 Numerical examples509

8.1 European option510

Consider a European option written on an underlying stock whose price S follows a jump diffusion511

process. Denote by ξ the random number representing the jump multiplier so that when a jump512

20

occurs, we have St = ξSt− . The risk neutral process followed by St is513

dStSt−

= (r − λκ)dt+ σdZ + d

( πt∑i=1

(ξi − 1)

)with κ = E[ξ]− 1 (8.1)

where E[·] denotes the expectation operator. Here, dZ is the increment of a Wiener process, r is514

the risk free rate, σ is the volatility, πt is a Poisson process with positive intensity parameter λ, and515

ξi are i.i.d. positive random variables. The density function f(y), y = log(ξ) is assumed double516

exponential (Kou and Wang, 2004)517

f(y) = pu η1 e−η1y1y≥0 + (1− pu) η2 e

η2y1y<0 (8.2)

with the expectation518

E[ξ] =pu η1

η1 − 1+

(1− pu) η2

η2 + 1. (8.3)

Given that a jump occurs, pu is the probability of an upward jump and (1− pu) is the probability519

of a downward jump.520

The price of a European call option v(x, τ) with x = logS is then given as the solution to521

vτ =σ2

2vxx + (r − σ2

2− λκ)vx − (r + λ)v + λ

∫ +∞

−∞v(x+ y)f(y) dy

with v(x, 0) = max(ex −K, 0) . (8.4)

The Green’s function for this problem is given in Appendix A.522

The particular parameters for this test are given in Table 8.1 with the results appearing in Table523

8.2. All methods obtain smooth second order convergence, with the exception of the FST/CONV524

Simpson rule method which gives fourth order convergence, due to the higher order quadrature525

method. This is to be expected in this case since there is a node at the strike. Increasing xmax, |xmin|526

altered results in the last 2 digits in the table. This is due to the effect of localizing the problem527

to [xmin, xmax], and the effects of FFT wrap-around.528

In order to stress these Fourier methods, we repeat this example, except now using an expiry529

time of T = .001. Since the Green’s function in the physical space converges to a delta function530

as T → 0, we can expect that this will be challenging for Fourier methods as a large number531

of terms will be required in the Fourier series in order to get an accurate representation of the532

Green’s function in the physical space. The results for this test are shown in Table 8.3. The533

monotone method with piecewise linear basis functions gives reasonable results for all grid sizes.534

The standard FST/CONV methods are quite poor, except for very large numbers of nodes. Indeed,535

using Simpson’s rule on coarse grids even results in values larger than S0 at S = S0 = 100, which536

violates the provable bound for a call option.537

This phenomenon can be explained by examining Figure 8.1, which shows the projection of538

the Green’s functions for the monotone method (piecewise linear basis function) and the truncated539

Green’s function for the FST/CONV method. The projection of the Green’s function for the540

monotone method in Figure 8.1(b) clearly has the expected properties: very peaked near x = 0 and541

non-negative for all x. In contrast, the FST/CONV numerical Green’s function is oscillatory and542

negative for some values of x. Figure 8.2 shows the FST/CONV (trapezoidal) solution compared to543

the Monotone (piecewise linear) solution, on a coarse grid with 512 nodes. The monotone solution544

can never produce a value less than zero (to within the tolerance). Note that monotonicity is clearly545

violated for the FST/CONV solution, with negative values for a call option. The oscillations are546

even more pronounced if Simpson’s quadrature is used for the FST/CONV method.547

548

21

Expiry time .25 yearsStrike K 100Payoff callInitial asset price S0 100Risk-free rate r .05Volatility σ .15λ .1η1 3.0465η2 3.0775pu 0.3445xmax log(S0) + 10xmin log(S0)− 10ε1, ε2 10−6

Asymptotic form x→∞ A(x) = ex

Table 8.1: European call option test.

Monotone Methods FST/CONV

Piecewise linear Piecewise constant Trapezoidal Simpson

N Value Ratio Value Ratio Value Ratio Value Ratio

29 3.9808516210 3.9443958729 3.9075619850 3.9784907318210 3.9753205007 3.9662547470 3.9571661688 3.9737010716211 3.9739391670 4.0 3.9716756819 4.0 3.9694107823 4.1 3.9734923202 23212 3.9735939225 4.0 3.9730282349 4.0 3.9724624589 4.0 3.9734796846 17213 3.9735076171 4.0 3.9733662066 4.0 3.9732247908 4.0 3.9734789013 16214 3.9734860412 4.0 3.9734506895 4.0 3.9734153372 4.0 3.9734788524 16

Table 8.2: European call option test: value at τ = 0, S = S0. Parameters in Table 8.1. N = numberof nodes. Ratio is the ratio of successive changes.

Remark 8.1 (Error in approximating equation (4.6) using equation (4.11)). An estimate of the549

error in computing the projected Green’s function is given in Appendix B, equation (B.5). We550

can see that a very small timestep effects the exponent in equation (B.5). For the extreme case of551

T = .001, N = 2056, problem in Table 8.1, we observe that for α = 8, then test2 in Algorithm 2552

is approximately 10−12, indicating a very high accuracy projection can be achieved under extreme553

situations. For the same problem (2056 nodes) with T = .25, we find that test2 in Algorithm 2 is554

approximately 10−16 for α = 2.555

556

From these tests we can conclude both that the monotone method is robust for all timestep sizes557

and that for smooth problems and large timesteps, the monotone method exhibits the expected558

slower rate of convergence compared to high order techniques.559

22

Monotone Methods FST/CONV

Piecewise linear Piecewise constant Trapezoidal Simpson

N Value Ratio Value Ratio Value Ratio Value Ratio

29 .19662316859 .94284763015 .24774086499 319.45747026210 .19467436458 .041410269769 .21909081933 521.62802838211 .19376651687 2.1 .15335986938 -8.0 .18611676723 .87 439.13444172 -2.5212 .19346709107 3.0 .18477993505 3.6 .17728640855 3.7 27.002978049 0.2213 .19339179620 4.0 .19127438852 4.8 .18913280108 -.75 .19367805822 15214 .19337297842 4.0 .19284673379 4.1 .19231903134 3.7 .19338110881 9× 104

Table 8.3: European call option test: value at τ = 0, S = S0. Parameters in Table 8.1 but T = .001.N = number of nodes. Ratio is the ratio of successive changes.

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(Gre

en

Fu

nctio

n)*

de

lx

(a) FST/CONV Green’s function, truncatedFourier series. Scaled by ∆x.

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(Gre

en

Fu

nctio

n)*

de

lx

(b) Monotone method, Green’s function pro-jected on linear basis functions.

Figure 8.1: European call option test: parameters in Table 8.1 but T = .001. FST/CONV methodtruncated Fourier series (N = 2048). Monotone method shows g(x,∆τ, α)∆x, with N = 2048, α = 4.

8.2 Bermudan option with non-proportional discrete dividends560

Let us now assume that we have the same underlying process (8.1) as in the previous subsection,561

except that the density function for y = log(ξ) is assumed normal562

f(y) =1√

2π γe− (y−ν)2

2γ2 (8.5)

with expectation E[ξ] = eν+γ2/2. Rather than a European option, we will now consider a Bermudan563

put option which can be early exercised at fixed monitoring times τn. In addition, the underlying564

asset pays a fixed dividend amount D at τ−n , that is, immediately after the early exercise oppor-565

tunity in forward time. Between monitoring dates, the option price is given by equation (8.4). At566

monitoring dates we have the condition567

v(x, τ+n ) = max(v(log(max(ex −D, exmin)), τ−n ), P (x)) with

P (x) = payoff = max(K − ex, 0) . (8.6)

23

50 55 60 65 70 75 80 85 90 95 100

Asset Price

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Op

tio

n V

alu

e

Monotone

Linear

FST/CONV

Trapezoidal

Figure 8.2: European call option test. Parameters in Table 8.1 but T = .001. N = 512. For themonotone solution, α = 4 (see Algorithm 2).

The expression max(ex −D, exmin) in equation (8.6) ensures that the no-arbitrage condition holds,568

that is, the dividend cannot be larger than the stock price, taking into account the localized grid.569

Linear interpolation is used to evaluate the option value in equation (8.6). The parameters for this570

problem are listed in Table 8.4 with the numerical results given in Table 8.5. All methods perform571

similarly, with second order convergence. We can see here that once we use a linear interpolation572

to impose the control, there is no benefit, in terms of convergence order, to using a high order573

method.574

9 Multiperiod mean variance optimal asset allocation problem575

In this section we give an example of a realistic problem with complex controls, the multiperiod576

mean variance optimal asset allocation problem. Here we consider the case of an investor with a577

portfolio consisting of a bond index and a stock index. The amount invested in the stock index578

follows the process under the objective measure579

dStSt−

= (µ− λκ)dt+ σdZ + d

( πt∑i=1

(ξi − 1)

)(9.1)

with the double exponential jump size distribution (8.2), while the amount in the bond index follows580

dBt = rBt dt . (9.2)

The investor injects cash qn at time time tn ∈ T with total wealth at time t being Wt = St+Bt.581

Let W−n = S−n +B−n be the total wealth before cash injection. It turns out that in the multiperiod582

mean variance case, in some circumstances, it is optimal to withdraw cash from the portfolio (Cui583

et al., 2014; Dang and Forsyth, 2016). Denote this optimal cash withdrawal as c∗n. The total wealth584

after cash injection and withdrawal is then585

W+n = W−n + qn − c∗n . (9.3)

24

Expiry time 10 yearsStrike K 100Payoff putInitial asset price S0 100Risk-free rate r .05Volatility σ .15Dividend D 1.00Monitoring frequency ∆τ 1.0 yearsλ .1ν -1.08γ .4xmax log(S0) + 10xmin log(S0)− 10ε1, ε2 10−6

Asymptotic form x→∞ A(x) = 0

Table 8.4: Bermudan put option test.

Monotone Methods FST/CONV

Piecewise linear Piecewise constant Trapezoidal Simpson

N Value Ratio Value Ratio Value Ratio Value Ratio

29 24.811127744 24.806532754 24.801967268 24.802639420210 24.789931363 24.788800257 24.787670043 24.787731820211 24.782264461 2.8 2.4.781982815 2.6 24.781701225 2.4 24.781787212 2.5212 24.781134292 6.8 24.781063962 7.4 24.780993635 8.4 24.781007785 7.6213 24.780822977 3.6 24.780805394 3.6 24.780787811 3.4 24.780788678 3.6214 24.780744620 4.0 24.780740225 4.0 24.780735831 4.0 24.780737159 4.3

Table 8.5: Bermudan put option test: value at τ = 0, S = S0. Parameters in Table 8.1. N =number of nodes. Ratio is the ratio of successive changes.

We then select an amount b∗n to invest in the bond, so that586

B+n = b∗n and S+

n = W+n − b∗n . (9.4)

Since only cash withdrawals are allowed we have c∗n ≥ 0. The control at rebalancing time tn consists587

of the pair (b∗n, c∗n). That is, after withdrawing c∗n from the portfolio we rebalance to a portfolio588

with S+n in stock and B+

n in bonds. A no-leverage and no-shorting constraint is enforced by589

0 ≤ b∗n ≤W+n . (9.5)

In order to determine the mean-variance optimal solution to this asset allocation problem, we make590

use of the embedding result (Li and Ng, 2000; Zhou and Li, 2000). The mean-variance optimal591

25

strategy can be posed as592

min{(b∗0,c∗0),...,(b∗M−1,c

∗M−1)}

E[(W ∗ −WT )2]

subject to

(St, Bt) follow processes (9.1), (9.2) ; t /∈ TW+n = S−n +B−n + qn − c∗n,

S+n = W+

n − b∗n, B+n = b∗n ; t ∈ T

0 ≤ b∗n ≤W+n

c∗n ≥ 0

, (9.6)

where W ∗ can viewed as a parameter which traces out the efficient frontier.593

Let594

Q` =M−1∑j=`+1

e−r(tj−t`)qj (9.7)

be the discounted future contributions to the portfolio at time t`. If595

(W−n + qn) > W ∗e−r(T−tn) −Qn , (9.8)

then the optimal strategy is to withdraw cash c∗n = W−n + qn − (W ∗e−r(T−tn) − Qn) from the596

portfolio, and invest the remainder(W ∗e−r(T−tn) − Qn

)in the risk free asset. This is optimal in597

this case since then E[(W ∗ −WT )2] = 0 (Cui et al., 2012; Dang and Forsyth, 2016), which is the598

minimum of problem (9.6).599

In the following we will refer to any cash withdrawn from the portfolio as a surplus or free cash600

flow (Bauerle and Grether, 2015). For the sake of discussion, we will assume that the surplus cash601

is invested in a risk-free asset, but does not contribute to the computation of the terminal mean602

and variance. Other possibilities are discussed in Dang and Forsyth (2016).603

The solution of problem (9.6) is the so-called pre-commitment solution. We can interpret the604

pre-commitment solution in the following way. At t = 0, we decide which Pareto point is desirable605

(that is, a point on the efficient frontier). This fixes the value of W ∗. At any time t > 0, we can606

regard the optimal policy as the time-consistent solution to the problem of minimizing the expected607

quadratic loss with respect to the fixed target wealth W ∗, which can be viewed as a useful practical608

objective function (Vigna, 2014; Menoncin and Vigna, 2017).609

9.1 Optimal control problem610

A brief overview of the PIDE for the solution of the mean-variance optimal control problem is given611

below (we refer the reader to Dang and Forsyth (2014) for additional details).612

Let the value function v(x,b,τ) with τ = T − t be defined as613

v(x, b, τ) = inf{(b∗0,c∗0),...,(b∗M ,c

∗M )}

{E

[(min(WT −W ∗, 0))2

∣∣∣∣ logS(t) = x,B(t) = b

]}. (9.9)

Let the set of observation times backward in time be T = {τ0, τ1, . . . , τM}. For τ /∈ T , v satisfies614

vτ = Lv + rbvb where

Lv ≡ σ2

2vxx + (µ− σ2

2− λκ)vx − (µ+ λ)v + λ

∫ ∞−∞

v(x+ y)f(y) dy

v(x, b, 0) = (min(ex + b−W ∗, 0))2 (9.10)

26

on the localized domain (x,b) ∈ [xmin, xmax]× [0, bmax].615

If g(x, τ) is the Green’s function of vτ = Lv then the solution of equation (9.10) at τ−n+1, given616

the solution at τ+n , τn ∈ T is617

v(x, b, τ−n+1) =

∫ xmax

xmin

g(x− x′,∆τ)v(x′, berb∆τ , τ+n ) with ∆τ = τn+1 − τn . (9.11)

Equation (9.11) can be regarded as a combination of a Green’s function step for the PIDE vτ = Lv618

and a characteristic technique to handle the rbvb term. At rebalancing times τn ∈ T ,619

v(x, b, τ+n ) = min

(b∗,c∗)v(x′, b∗, τ−n )

subject to

c∗ = max(ex + b+ qM−n −QM−n, 0)

W ′ = ex + b+ qM−n − c∗

0 ≤ b∗ ≤W ′

x′ = log(max(W ′ − b∗, exmin)

) (9.12)

where Q` is defined in equation (9.7).620

9.2 Computational details621

We solve problem (9.9) combined with the optimal control (9.12) on the localized domain (x,b) ∈622

[xmin, xmax]× [0, bmax]. We discretize in the x direction using an equally spaced grid with Nx nodes623

and an unequally spaced grid in the B direction with Nb nodes. Set Bmax = exmax and denote the624

discrete solution at (xm, bj , τ+n ) by625

(vnm,j)+ = v(xm, bj , τ

+n )

(vn)+ = {(vnm,j)+}m=−Nx/2,...,Nx/2−1;j=1,...,Nb

(vnj )+ = [(vn−Nx/2,j)+, . . . , (vnNx/2−1,j)

+]. (9.13)

Let I∆x,∆b(x, b)(vn)− be a two dimensional linear interpolation operator acting on the discrete626

solution values (vn)−. Given the solution at τ+n , we use Algorithm 3 to advance the solution to627

τ−n+1. For the mean variance problem, we extend this algorithm to approximate equation (9.11),628

which is described in Algorithm 4.629

Algorithm 4 Advance time (vn)+ → (vn+1)−.

Require: (vn)+ ; G = {G(ωm,∆τ, α)}, m = −Nx/2, . . . , Nx/2− 1 (from Algorithm 2)1: for j = 1, . . . , Nb do {Advance time loop}2: vintm,j = I∆x,∆b(xm, bje

r∆τ )(vn)+ ; m = −Nx/2, . . . , Nx/2− 1

3: V = FFT [ vintj ]

4: (vn+1j )− = iFFT [ V ◦ G ] {iFFT( Hadamard product )}

5: end for{End advance time loop}

In order to advance the solution from τ−n+1 to τ+n+1, we approximate the solution to the optimal630

control problem (9.12). The optimal control is approximated by discretizing the candidate control631

27

b∗ using the discretized b grid and exhaustive search:632

v(xm, bj , τ+n ) = min

(b∗,c∗)I∆x,∆b((x

∗, b∗)(vn+1)−

subject to

c∗ = max(exm + bj + qM−n −QM−n, 0)

W ′ = exm + bj + qM−n − c∗

b∗ ∈ {b1, . . . ,min(bmax,W′)}

x∗ = log(max(W ′ − b∗, exmin)

) . (9.14)

This is a convergent algorithm to the solution of the original control problem as Nx, Nb →∞. This633

can be proved using similar steps as in the finite difference case (Dang and Forsyth, 2014). For634

brevity we omit the proof.635

Using the control determined from solving problem (9.9), we can determine E[WT ] and std[WT ]636

by solving an additional linear PIDE, see (Dang and Forsyth, 2014) for details.637

Remark 9.1 (Practical implementation enhancements). As noted by several authors, since the638

Green’s function and the solution is real, the Fourier coefficients satisfy symmetry relations. Hence639

G(ωk,∆τ, α) and V need to be computed and stored only for ωk ≥ 0. It is also possible to arrange640

the step in line 2 of Algorithm 4 and the optimal control step of (9.14) so that only a single641

interpolation error is introduced at each node. Note that the Fourier series representation of the642

Green’s function is only used to compute the projection of the Green’s function onto linear basis643

functions. After this initial step, we use FFTs only to efficiently carry out a dense matrix-vector644

multiply (the convolution) at each step. Use of the FFT here is algebraically identical to carrying645

out the convolution in the physical space. The only approximation being used in this step is the646

periodic extension of the solution.647

9.3 Numerical example648

The data for this problem is given in Table 9.1. The data was determined by fitting to the monthly649

returns from the Center for Research in Security Prices (CRSP) through Wharton Research Data650

Services, for the period 1926:1- 2015:12.3 We use the monthly CRSP value-weighted (capitalization651

weighted) total return index (“vwretd”), which includes all distributions for all domestic stocks652

trading on major US exchanges, and the monthly 90-day Treasury bill return index from CRSP.653

Both this index and the equity index are in nominal terms, so we adjust them for inflation by654

using the US CPI index (also supplied by CRSP). We use real indexes since investors saving for655

retirement are focused on real (not nominal) wealth goals.656

As a first test, we fix W ∗ = 1022, and then increase the number nodes in the x direction (Nx)657

and in the b direction (Nb). We use the monotone scheme, with linear basis functions. In Table 9.2,658

we show the value function v(0, 0,T ) and the mean E[WT ] and standard deviation std[WT ] of the659

final wealth, which are of practical importance. The value function shows smooth second order660

convergence, which is to be expected. Even though the optimal control is correct only to order ∆b661

(since we optimize by discretizing the controls and using exhaustive search), the value function is662

correct to O(∆b)2 (since it is an extreme point).663

We expect that the derived quantities E[WT ], std[WT ], which are based on the controls com-664

puted as a byproduct of computing the value function, should show a lower order convergence.665

3More specifically, results presented here were calculated based on data from Historical Indexes, c©2015 Center forResearch in Security Prices (CRSP), The University of Chicago Booth School of Business. Wharton Research DataServices was used in preparing this article. This service and the data available thereon constitute valuable intellectualproperty and trade secrets of WRDS and/or its third-party suppliers.

28

Expiry time T 30 yearsInitial wealth 0Rebalancing frequency yearlyCash injection {qi}i=0,...,29 10Real interest rate r .00827Volatility σ .14777µ .08885λ .3222η1 4.4273η2 5.262pu 0.2758xmax log(100) + 5xmin log(100)− 10ε1, ε2 10−6

Asymptotic form E[(WT −W ∗)2], x→∞ A(x) = 0

Table 9.1: Multiperiod mean variance example. Parameters determined by fitting to the real (infla-tion adjusted) CRSP data for the period 1926:1-2015:12. Interest rate is the average real return on 90day T-bills.

Nx Nb Value function Ratio E[WT ] Ratio std[WT ] Ratio

512 305 97148.899100 N/A 824.02599269 N/A 240.73884508 N/A1024 609 97042.740997 N/A 824.07104985 N/A 240.55534019 N/A2048 1217 97014.471301 3.8 824.09034690 2.3 240.51245396 4.34096 2433 97007.286530 3.9 824.08961667 -26 240.49691620 2.78192 4865 97005.451814 3.9 824.09295889 -.22 240.49585213 14.6

Table 9.2: Test of convergence of optimal multiperiod mean variance investment strategy. Monotonemethod, linear basis functions. Parameters in Table 9.1. Fixed W ∗ = 1022. Ratio is the ratio ofsuccessive changes.

Recall that these quantities are evaluated by storing the controls and then solving a linear PIDE.666

In fact we do see somewhat erratic convergence for these quantities. As an independent check, we667

used the stored controls from solving for the value function (on the finest grid), and then carried668

out Monte Carlo simulations to directly compute the mean and standard deviation of the final669

wealth. The results are shown in Table 9.3.670

Of more practical interest is the following computation. In Table 9.4 we show the results671

obtained by rebalancing to a constant weight in equities at each monitoring date. We specify that672

the portfolio is rebalanced to .60 in stocks and .40 in bonds (a common default recommendation).673

We then solve for the value function using the monotone Fourier method, allowing W ∗ to vary, but674

fixing the expected value so that E[WT ] is the same as for the 60 : 40 constant proportion strategy.675

This is done by using a Newton iteration, where each evaluation of the residual function requires a676

solve for the value function and the expected value equation. The results of this test are shown in677

Table 9.5. In this case, fixing the mean and allowing W ∗ to vary, results in smooth convergence of678

the standard deviation. From a practical point of view, we can see that the optimal strategy has679

the same expected value as the constant proportion strategy, but the standard deviation is reduced680

29

Nsim E[WT ] std[WT ]

1.6× 105 824.3425 (1.55) 240.22636.4× 105 823.6719 (0.78) 240.72782.56× 106 824.0077 (0.39) 240.43361.024× 107 824.1043 (0.19) 240.5217

Table 9.3: Monte Carlo simulation results, based on optimal controls from solving for the valuefunction using the monotone Fourier technique. Numbers in brackets are the standard error, 99%confidence level, for the mean. Compare with Table 9.2. Parameters in Table 9.1. Fixed W ∗ = 1022.

E[WT ] std[WT ] Median[WT ]

824.10047 511.8482 704

Table 9.4: Portfolio rebalanced to .60 in stocks and .40 in bonds at each monitoring date. Closedform expression for mean and standard deviation. Median computed using Monte Carlo simulation.Parameters in Table 9.1.

from 512 to 241, and the median of the optimal strategy is 936 compared to a median of 704 for681

the constant proportion strategy. A heat map of the optimal strategy is shown in Figure 9.1.682

10 Conclusions683

Many problems in finance give rise to discretely monitored complex control problems. In many684

cases, the optimal controls are not of a simple bang-bang type. It then happens that a numerical685

procedure must be used to determine the optimal control at discrete points in the physical domain.686

In these situations, there is little hope of obtaining a high order accurate solution, after the control687

is applied. If we desire a monotone scheme, which increases robustness and reliability for our688

computations, then we are limited to the use of linear interpolation, hence we can get at most689

second order accuracy.690

Traditional FST/CONV methods assume knowledge of the Fourier transform of the Green’s691

function but then approximate this function by a truncated Fourier series. As a result these methods692

are not monotone. Instead when the Fourier transform of the Green’s function is known, then we693

carry out a pre-processing step by projecting the Green’s function (in the physical space) onto a set694

Nx Nb E[WT ] std[WT ] Ratio

512 305 824.10047 240.79440842 N/A1024 609 824.10047 240.57925928 N/A2048 1217 824.10047 240.52022512 3.64096 2433 824.10047 240.50571976 4.18192 4865 824.10047 240.50220544 4.1

Table 9.5: At each refinement level W ∗ is determined so that E[WT ] = 824.10047. The median onthe finest grid is computed by storing the controls and using Monte Carlo simulation. Median[WT ] =936. Ratio is the ratio of successive changes. Parameters in Table 9.1.

30

Figure 9.1: Optimal strategy, fraction of portfolio invested in stock, as a function of current totalreal wealth Wt = St +Bt and forward time t. Parameters in Table 9.1.

of linear basis functions. These integrals can then be computed to within a specified tolerance and695

this allows us to guarantee a monotone scheme to within the tolerance. This monotone scheme is696

robust to small timesteps, which is observably not the case for the standard FST/CONV methods,697

and indeed is a major pitfall of the latter methods.698

When the Green’s function depends on time only through the timestep size and the monitoring699

dates for the control are equally spaced (which is typically the case), then the final monotone700

algorithm has the same complexity per step as the original FST/CONV algorithms, and the same701

order of convergence for smooth control problems. It is a simple process to add this preprocessing702

step to existing FST/CONV software. This results in more robust and more reliable algorithms for703

optimal stochastic control problems.704

11 Acknowledgements and Declaration of Interest705

This work was supported by the Natural Sciences and Engineering Research Council of Canada706

(NSERC). The authors report no conflicts of interest. The authors alone are responsible for the707

content and writing of the paper.708

Appendices709

A Green’s functions710

Consider the PIDE711

vτ =σ2

2vxx + (µ− σ2

2− λκ)vx − (ρ+ λ)v + λ

∫ +∞

−∞v(x+ y)f(y) dy . (A.1)

31

If, for example, ρ = µ = r where r is the risk-free rate, then this is the option pricing equation,712

while if ρ = 0 then the right hand side of equation (A.1) is Lv in the mean variance case.713

Let714

v(ω, τ) =

∫ ∞−∞

V (ω, τ)e2πiωx dω

f(ω, τ) =

∫ ∞−∞

F (ω, τ)e2πiωx dω . (A.2)

Substituting equation (A.2) into equation (A.1) gives715

V (ω, τ)τ = Ψ(ω)V (ω, τ) where (A.3)

Ψ(ω) =

(−σ

2

2(2πω)2 + (µ− λκ− σ2

2)(2πiω)− (ρ+ λ) + λF (ω)

),

with F (ω) being the complex conjugate of F (ω). Integrating equation (A.3) gives716

V (ω, τ + ∆τ) = eΨ(ω)∆τV (ω, τ) ,

from which we can deduce that the Fourier transform of the Green’s function G(ω,∆τ) is717

G(ω,∆τ) = eΨ(ω)∆τ . (A.4)

In the case of a double exponential jump distribution (8.2), then718

F (ω) =pu

1− 2πiω/η1+

1− pu1 + 2πiω/η2

while in the case of a log-normal jump size distribution (8.5)719

F (ω) = e2(πiων−(πωγ)2) .

From equation (A.3) and (A.4) we have720

G(0,∆τ) = e−ρ∆τ ,

which means that in these cases C1 =∫R g(x,∆τ) dx is721

C1 =

{e−r∆τ option pricing

1 mean variance asset allocation.

B Convergence of truncated Fourier series for the projected Green’s722

functions.723

Since the Green’s function for equation (A.1) is a smooth function for any finite ∆τ , we can expect724

uniform convergence of the Fourier series to the exact Green’s function, assuming that σ > 0. This725

can also be seen from the exponential decay of the Fourier coefficients, which we demonstrate in726

this Appendix. Since the exact Green’s function is non-negative, the projected Green’s function727

(4.8) then converges to a non-negative value at every point yj . Consider the case of the truncated728

projection on linear basis functions729

g(yj ,∆τ, α) =1

P

αN/2−1∑k=−αN/2

e2πiωkyj

(sin2 πωk∆x

(πωk∆x)2

)G(ωk,∆τ) with ωk =

k

P

32

and ∆x = PN . The error in the truncated series is then730

|g(yj ,∆τ, α)− g(yj ,∆τ,∞)| =

∣∣∣∣ 1

P

∞∑k=αN/2

e2πiωkyj

(sin2 πωk∆x

(πωk∆x)2

)G(ωk,∆τ)

+1

P

−αN/2−1∑k=−∞

e2πiωkyj

(sin2 πωk∆x

(πωk∆x)2

)G(ωk,∆τ)

∣∣∣∣≤ 2

P

∞∑k=αN/2

1

(πωk∆x)2|G(ωk,∆τ)|

≤ 2

P· 4

π2α2

∞∑k=αN/2

|G(ωk,∆τ)| . (B.1)

Noting that Re(F (ω)) ≤ 1, we then have731

Re(Ψ(ω)) = −σ2(2πω)2

2− (ρ+ λ) + λRe(F (ω))

≤ −σ2(2πω)2

2− (ρ+ λ) + λ ≤ −σ

2(2πω)2

2

since ρ ≥ 0. Hence732

|G(ω,∆τ)| = |eΨ(ω)∆τ | ≤ e−σ2(2πω)2∆τ

2 . (B.2)

If we let C4 = 2σ2π2∆τP 2 then equations (B.2) and (B.1) implies733

|g(yj ,∆τ, α)− g(yj ,∆τ,∞)| ≤ 8

Pπ2α2

∞∑k=αN/2

e−C4k2.

Bounding the sum gives734

|g(yj ,∆τ, α)− g(yj ,∆τ,∞)| ≤ 8

Pπ2α2· e−C4N2α2/4

1− e−C4Nα. (B.3)

Consider the monotonicity test in Algorithm 2, line 5, given by735

test1 =∑j

∆xmin(g(yj ,∆τ, α), 0).

Noting that g(yj ,∆τ,∞) ≥ 0, and from equation (B.3) and∑

j ∆x = P , we have736

|test1| ≤8

π2α2· e−C4N2α2/4

1− e−C4Nα, (B.4)

so that usually this test is satisfied to within round off for α = 2, 4.737

Consider now the accuracy test on line 6 of Algorithm 2, given by738

test2 = maxj

∆x|g(yj ,∆τ, α)− g(yj ,∆τ, α/2)|

33

which we see is bounded by739

|test2| ≤ ∆xmaxj

(|g(yj ,∆τ, α)− g(yj ,∆τ,∞)|+ |g(yj ,∆τ, α/2)− g(yj ,∆τ,∞)|

)≤ ∆xmax

j

(2|g(yj ,∆τ, α/2)− g(yj ,∆τ,∞)|

)≤ 64

π2α2· ∆x

P· e−C4N2α2/16

1− e−C4Nα/2. (B.5)

This test will also be satisfied for small values of α, although this will require larger values of α740

than for the monotonicity test (B.4).741

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